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Theory of numerical range and numerical radius for tensors is not studied much in the literature. In 2016, Ke {\it et al.} [Linear Algebra Appl., 508 (2016) 100-132] introduced first the notion of numerical range of a tensor via the $k$-mode product. However, the convexity of the numerical range via the $k$-mode product was not proved by them. In this paper, the notion of numerical range and numerical radius for even-order square tensors using inner product via the Einstein product are introduced first. We provide some sufficient conditions using numerical radius for a tensor to being unitary. The convexity of the numerical range is also proved. We also provide an algorithm to plot the numerical range of a tensor. Furthermore, some properties of the numerical range for the Moore--Penrose inverse of a tensor are discussed., Comment: 20 pages

In this paper, we provide a few properties of the weighted Moore--Penrose inverse for an arbitrary order tensor via the Einstein product. We again obtain some new sufficient conditions for the reverse-order law of the weighted Moore--Penrose inverse for even-order square tensors. We then present several characterizations of the reverse-order law for tensors of arbitrary order., Comment: 15 pages

The equality $(\mc{A}\n\mc{B})^\dagger = \mc{B}^\dagger \n \mc{A}^\dagger$ for any two complex tensors $\mc{A}$ and $\mc{B}$ of arbitrary order, is called as the {\it reverse-order law} for the Moore-Penrose inverse of arbitrary order tensors via the Einstein product. Panigrahy {\it et al.} [Linear Multilinear Algebra; 68 (2020), 246-264.] obtained several necessary and sufficient conditions to hold the reverse-order law for the Moore-Penrose inverse of even-order tensors via the Einstein product, very recently. This notion is revisited here among other results. In this context, we present several new characterizations of the reverse-order law of arbitrary order tensors via the same product. More importantly, we illustrate a result on the Moore-Penrose inverse of a sum of two tensors as an application of the reverse-order law which leaves an open problem. We recall the definition of the Frobenius norm and the spectral norm to illustrate a result for finding the additive perturbation bounds of the Moore-Penrose inverse under the Frobenius norm. We conclude our paper with the introduction of the notion of sub-proper splitting for tensors which may help to find an iterative solution of a tensor multilinear system.

In this paper, we first give an expression for the Moore-Penrose inverse of the product of two tensors via the Einstein product. We then introduce a new generalized inverse of a tensor called product Moore-Penrose inverse. A necessary and sufficient condition for the coincidence of the Moore-Penrose inverse and the product Moore-Penrose inverse is also proposed. Finally, the triple reverse order law of tensors is introduced.

Reverse order law for the Moore-Penrose inverses of tensors are useful in the field of multilinear algebra. In this paper, we first prove some more identities involving the Moore-Penrose inverse of tensors. We then obtain a few necessary and sufficient conditions for the reverse order law for the Moore-Penrose inverse of tensors via the Einstein product.